Let me take a stab at this.
Verticutting is always a good thing to do at least once per year, as is aeration.
However without knowing what type of grass it is, it's hard to say exactly what your problem is. I'm not real familiar with the turf varieties in KC, so I don't know what to expect. If your grass is bermuda, it is supposed to be brown this time of year, as it is dormant. The only way you could make it green is to actually overseed it in the fall before the temps get too cool.
Is it possible that your neighbor is overseeding in the fall, or is he making regular fertilizer applications and you are not? If the ground is dry, that is a definite sign you are not giving it enough water.
I can't imagine what he would be using chemical wise that has an odor, none of the growth regulators give off much of a scent, and I doubt he would know how to use and apply correctly.
I'm guessing it is some kind of fertilizer or conditioner, and just about any of them are going to be harmless to pets. Just stay away from the ones coated with pendimethalin or any other weed control.
Check with a local nursery, bring them a sample of the grass so you can get it indentified. They should be able to help.